Does a quantum computer resolve the halting problem and would that advance strong AI?
It depends a bit on what you mean by 'quantum computer'. The 'conventional' notion is that quantum computation buys a (in some cases, exponential) speedup - it doesn't change what can be computed, just how quickly.

In contrast, advocates of hypercomputation claim that quantum effects may make it possible to do infinite computations in finite time. Note, however, that this is not a mainstream belief - the reknowned logician Martin Davis has written an article claiming that hypercomputation is a myth.

Roger Penrose has also claimed that quantum vibrations in neural microtubules may be responsible for consciousness.